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Attractors for the stochastic 3D Navier-Stokes equations. (English) Zbl 1059.35100
Summary: In [J. Nonlinear Sci. 7, 475–502 (1997; Zbl 0903.58020)] J. M. Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3D Navier-Stokes equations have a global attractor provided that all weak solutions are continuous from $\left(0,\infty \right)$ into ${L}^{2}$. In this paper we adapt his framework to treat stochastic equations: we introduce a notion of a stochastic generalised semiflow, and then show a similar result to Ball’s concerning the attractor of the stochastic 3D Navier-Stokes equations with additive white noise.
##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35R60 PDEs with randomness, stochastic PDE 37L30 Attractors and their dimensions, Lyapunov exponents 60H15 Stochastic partial differential equations 35B41 Attractors (PDE) 76D05 Navier-Stokes equations (fluid dynamics) 76M35 Stochastic analysis (fluid mechanics)